In this section we prove Theorem REF . The tools we will employ are essentially taken from linear algebra. The same kind of ideas has been applied in [1]} to reformulate problems of intersection theory for paths in the Lagrange Grassmannian in terms of finite dimensional linear algebra. The classical approach to second order minimality conditions rephrases the problem of counting the number of negative eigenvalues of the Second Variation (the counter part of our quadratic form \(\mathcal {J}\) ) as counting the intersection number of a curve in a finite-dimensional manifolds, known as the Lagrangian Grassmannian, with an appropriate submanifold. For further references on the continuum approach see for instance [2]}, [3]}, [4]} and reference therein.
